Taking the inverse sine of a number >1

[Special K]

If I do sin^-1 1.1 on my TI86, I get the complex number (1.57079632679,-.443568254385i). What does this mean? How did it come up with that answer? How could you construct a triangle in which the ratio between the opposite side and hypotenuse is greater than 1?

 

[TheCorporal]

Think Imaginary numbers. Thats the only way the calculator can make it work. These i numbers are represented as 2 coordinates, and is the reason there is an "i" at the end of the last coordinate. I wish I had studied harder during Calc or I would be able to explain why its represented as those coordinates. I think I remember that there is some special way of representing i numbers.

Hopefully im right about this, and hope this helps...

 

[samgau]

If i remember correctly, the i value can be plotted on the Y axis and the numerical value goes on the X axis comparing to a regular x-y coordinate system

 

[m0ti]

It's because sin is defined as (exp(ix) - exp (-ix))/2i where i is the root of -1. exp means e to the power of.

And sin is only defined between -1 and 1 on the REAL numbers. so it should come as no surprise that for complex numbers, the values can be over this.

 

[Special K]

Looks like some things I will learn in calc I this fall. I never really worked with e last year in precalc, basically the teacher just told us what it was. I have never seen sine defined the way that m0ti showed it.

 

[TheCorporal]

Um, the variable you are working with is i, not e. e is a constant and is used in logs and natural logs I believe. You know, stuff to help solve exponentials and some derivatives and integrations.

 

[RossGr]

Neither i nor e is a variable. Both are well defined constants,
i = Sqrt(-1)
e= 2.7???.... is the base of the natural logs.
x is the variable.

The definition for sine given earlier in this thread it more general then the standard definition presented in HS. The exponitial defintion is very useful when generating solutions to differential equations. The same general solution can result in either sinusoidal or exponential functions depenting on the conditions present in the original problem.

 

[TheCorporal]

Heh I dint mean variables, he just reffered to e as a variable and then I just said i was one as well. Heh. Anyways, imaginary numbers were fun =)

 

[Special K]

Well I knew it was a constant, I guess I just didn't word my post correctly